Average Error: 0.0 → 0.0
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(1 - v \cdot v\right) \cdot \left(\left(\sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}} \cdot \sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{2}}{4}\right)\]

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{\color{blue}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \left(1 - v \cdot v\right)\]

  4. Applied sqrt-prod0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\left(\sqrt{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}} \cdot \sqrt{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}\right)}\right) \cdot \left(1 - v \cdot v\right)\]

  5. Final simplification0.0

    \[\leadsto \left(1 - v \cdot v\right) \cdot \left(\left(\sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}} \cdot \sqrt{\sqrt{1 - \left(v \cdot v\right) \cdot 3}}\right) \cdot \frac{\sqrt{2}}{4}\right)\]

Reproduce

herbie shell --seed 2018365 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))

Details

Time bar (total: 1.1m)Debug log

start159.0ms

Algorithm
intervals

setup489.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize40.0ms

Local error

Found 4 expressions with local error:

0.2b
(* 3 (* v v))
0.0b
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
0.0b
(sqrt (- 1 (* 3 (* v v))))
0.0b
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))

rewrite27.0ms

Algorithm
rewrite-expression-head
Counts
4 → 88
Calls

4 calls. Slowest were:

13.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
10.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
1.0ms
(* 3 (* v v))

series306.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

119.0ms
(* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v)))
103.0ms
(* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v)))))
42.0ms
(sqrt (- 1 (* 3 (* v v))))
42.0ms
(* 3 (* v v))

simplify14.0s

Counts
78 → 100
Calls

78 calls. Slowest were:

829.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (sqrt (- 1 (* 3 (* v v)))) (sqrt (- 1 (* 3 (* v v))))) (sqrt (- 1 (* 3 (* v v))))))
595.0ms
(* (- (* v v)) (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))))
499.0ms
(* (* 4 (sqrt (+ 1 (* 3 (* v v))))) (+ (* 1 1) (+ (* (* v v) (* v v)) (* 1 (* v v)))))

prune1.7s

Pruning

3 alts after pruning (3 fresh and 0 done)

Merged error: 0b

localize23.0ms

Local error

Found 4 expressions with local error:

0.2b
(* 3 (* v v))
0.2b
(* 3 (* v v))
0.0b
(* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))
0.0b
(* (/ (sqrt 2) 4) (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))))

rewrite39.0ms

Algorithm
rewrite-expression-head
Counts
4 → 98
Calls

4 calls. Slowest were:

21.0ms
(* (/ (sqrt 2) 4) (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))))
13.0ms
(* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))
2.0ms
(* 3 (* v v))

series143.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

72.0ms
(* (/ (sqrt 2) 4) (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))))
38.0ms
(* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))
19.0ms
(* 3 (* v v))
13.0ms
(* 3 (* v v))

simplify11.2s

Counts
87 → 110
Calls

87 calls. Slowest were:

907.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))) (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))) (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))))
572.0ms
(* (* (* (/ (sqrt 2) 4) (/ (sqrt 2) 4)) (/ (sqrt 2) 4)) (* (* (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))) (* (* (sqrt (sqrt (- 1 (* 3 (* v v))))) (sqrt (sqrt (- 1 (* 3 (* v v)))))) (sqrt (sqrt (- 1 (* 3 (* v v))))))))
350.0ms
(- (* 1/4 (sqrt 2)) (+ (* 9/32 (* (sqrt 2) (pow v 4))) (* 3/8 (* (sqrt 2) (pow v 2)))))

prune2.1s

Pruning

3 alts after pruning (2 fresh and 1 done)

Merged error: 0b

localize69.0ms

Local error

Found 4 expressions with local error:

0.2b
(* 3 (* v v))
0.1b
(* (* 3 v) v)
0.1b
(* (pow v 4) -9)
0.0b
(/ (* (sqrt (sqrt (- 1 (* (* 3 v) v)))) (* (sqrt (sqrt (+ (* (pow v 4) -9) 1))) (sqrt 2))) (* 4 (sqrt (sqrt (+ 1 (* 3 (* v v)))))))

rewrite43.0ms

Algorithm
rewrite-expression-head
Counts
4 → 58
Calls

4 calls. Slowest were:

31.0ms
(/ (* (sqrt (sqrt (- 1 (* (* 3 v) v)))) (* (sqrt (sqrt (+ (* (pow v 4) -9) 1))) (sqrt 2))) (* 4 (sqrt (sqrt (+ 1 (* 3 (* v v)))))))
4.0ms
(* (* 3 v) v)
4.0ms
(* (pow v 4) -9)

series245.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

181.0ms
(/ (* (sqrt (sqrt (- 1 (* (* 3 v) v)))) (* (sqrt (sqrt (+ (* (pow v 4) -9) 1))) (sqrt 2))) (* 4 (sqrt (sqrt (+ 1 (* 3 (* v v)))))))
24.0ms
(* (pow v 4) -9)
23.0ms
(* (* 3 v) v)
17.0ms
(* 3 (* v v))

simplify6.7s

Counts
34 → 70
Calls

34 calls. Slowest were:

1.3s
(/ (* (* (* (sqrt (sqrt (- 1 (* (* 3 v) v)))) (sqrt (sqrt (- 1 (* (* 3 v) v))))) (sqrt (sqrt (- 1 (* (* 3 v) v))))) (* (* (* (sqrt (sqrt (+ (* (pow v 4) -9) 1))) (sqrt (sqrt (+ (* (pow v 4) -9) 1)))) (sqrt (sqrt (+ (* (pow v 4) -9) 1)))) (* (* (sqrt 2) (sqrt 2)) (sqrt 2)))) (* (* (* 4 (sqrt (sqrt (+ 1 (* 3 (* v v)))))) (* 4 (sqrt (sqrt (+ 1 (* 3 (* v v))))))) (* 4 (sqrt (sqrt (+ 1 (* 3 (* v v))))))))
523.0ms
(* (sqrt (sqrt (- (* 1 1) (* (* (* 3 v) v) (* (* 3 v) v))))) (* (sqrt (sqrt (- (* (* (pow v 4) -9) (* (pow v 4) -9)) (* 1 1)))) (sqrt 2)))
516.0ms
(/ (* (* (* (sqrt (sqrt (- 1 (* (* 3 v) v)))) (sqrt (sqrt (- 1 (* (* 3 v) v))))) (sqrt (sqrt (- 1 (* (* 3 v) v))))) (* (* (* (sqrt (sqrt (+ (* (pow v 4) -9) 1))) (sqrt 2)) (* (sqrt (sqrt (+ (* (pow v 4) -9) 1))) (sqrt 2))) (* (sqrt (sqrt (+ (* (pow v 4) -9) 1))) (sqrt 2)))) (* (* (* 4 (sqrt (sqrt (+ 1 (* 3 (* v v)))))) (* 4 (sqrt (sqrt (+ 1 (* 3 (* v v))))))) (* 4 (sqrt (sqrt (+ 1 (* 3 (* v v))))))))

prune1.8s

Pruning

3 alts after pruning (1 fresh and 2 done)

Merged error: 0b

localize22.0ms

Local error

Found 4 expressions with local error:

1.0b
(* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v))))))
0.2b
(* 3 (* v v))
0.0b
(sqrt (- 1 (* 3 (* v v))))
0.0b
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))

rewrite102.0ms

Algorithm
rewrite-expression-head
Counts
4 → 118
Calls

4 calls. Slowest were:

65.0ms
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))
27.0ms
(* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v))))))
2.0ms
(* 3 (* v v))

series219.0ms

Counts
4 → 12
Calls

4 calls. Slowest were:

91.0ms
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))
75.0ms
(* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v))))))
35.0ms
(sqrt (- 1 (* 3 (* v v))))
19.0ms
(* 3 (* v v))

simplify24.9s

Counts
123 → 130
Calls

123 calls. Slowest were:

720.0ms
(* (- (* v v)) (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))))
583.0ms
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))
568.0ms
(* (* (sqrt (/ (sqrt 2) 4)) (* (sqrt (/ (sqrt 2) 4)) (sqrt (- 1 (* 3 (* v v)))))) (- 1 (* v v)))

prune2.3s

Pruning

3 alts after pruning (0 fresh and 3 done)

Merged error: 0b

regimes124.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch5.0ms